reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;
reserve T,L1 for Sequence,
  O,O1,O2,O3,C for Ordinal;

theorem Th28:
  for d be BiFunction of A,L for q being QuadrSeq of d holds O <>
0 & O is limit_ordinal & dom T = O & (for O1 being Ordinal st O1 in O holds T.
  O1 = ConsecutiveDelta(q,O1)) implies ConsecutiveDelta(q,O) = union rng T
proof
  deffunc D(Ordinal,Sequence) = union rng $2;
  let d be BiFunction of A,L;
  let q be QuadrSeq of d;
  deffunc C(Ordinal,set) = new_bi_fun(BiFun($2,ConsecutiveSet(A,$1),L),Quadr(q
  ,$1));
  deffunc F(Ordinal) = ConsecutiveDelta(q,$1);
  assume that
A1: O <> 0 & O is limit_ordinal and
A2: dom T = O and
A3: for O1 being Ordinal st O1 in O holds T.O1 = F(O1);
A4: for O being Ordinal, It being object holds It = F(O) iff ex L0 being
Sequence st It = last L0 & dom L0 = succ O & L0.0 = d & (for C being Ordinal
st succ C in succ O holds L0.succ C = C(C,L0.C)) & for C being Ordinal st C in
  succ O & C <> 0 & C is limit_ordinal holds L0.C = D(C,L0|C) by Def16;
  thus F(O) = D(O,T) from ORDINAL2:sch 10(A4,A1,A2,A3);
end;
